ORBIT DETERMINATION OF LEO SATELLITES FOR A SINGLE PASS THROUGH A RADAR: COMPARISON OF METHODS

Transcription

1 ORBIT DETERMINATION OF LEO SATELLITES FOR A SINGLE PASS THROUGH A RADAR: COMPARISON OF METHODS Zakhary N. Khutorovsky, Sergey Yu. Kamensky and, Nikolay N. Sbytov Vympel Corporation, Moscow, Russia Kyle T. Alfriend Texas A&M University, USA

2 OUTLINE Introduction Least squares algorithm Recurrent algorithm Guarantee approach method

3 Introduction Initial Orbit Determination (IOD) Russia radar sites track each object and report the state vector and covariance matrix as the observation instead of observations. Current IOD approaches were established in the 1960s and 70s. Very little interest until recently. The planned radar fence upgrade and WFOV cameras will grow the catalog from the current objects to an estimated objects. Correlating tracks from these objects will be a major challenge. The accuracy of collisiom probability determination depend on the accuracy state and covariances. Need improved state estimates and covariances from the IOD.

4 The least squares algorithm The least squares method search for the minimum of ( ) Ψ(a) = n 1 (d σd 2 k d k (a)) (α σα 2 k α k (a)) (β k k σβ 2 k β k (a)) 2, k where k=1 d k,α k,β k k-th measurement range, azimuth, elevation (k =1, 2,..., n); t k epochofthek-thmeasurement; σ dk,σ αk,σ βk RMS of the errors d k,α k,β k ; a = a ( t) - state at epoch t. h k (a) =(d k (a), α k (a), β k (a)) values of the parameters of the k-th measurement, calculated using the vector a. ( ) Ψ The solution a min is obtained from a (a min) ( ) 1 2 Ψ and covariance K from K =0.5 a (a min). 2 =0

5 Recurrent algorithm The estimate a min of the least squares method has recurrent structure: the estimate of parameters based on k measurements can be written as a function of the same estimate for k 1 measurements and the k-th measurement. The recurrence feature of the least squares estimate can be extanded to more general case, when the parameters a k =a(t k ) of the system for the time t k (in our case the orbital parameters for the time t k ) in addition to deterministic component have a random one which has Markov s feature (the future for the fixed past depends only on the present ), i.e. p (a k a k 1, a k 2,...) =p (a k a k 1 ) (M) where p (a k...) - conditional probability density a k. The condition (M) is satisfied if a k = f k (a k 1 ) + γ k (M1) where γ k - noise of the system - time independent random perturbations with zero mean and covariation matrices Γ k.

6 For (M), (M1) and independent errors of the measurements for the aposteriori (after acquisition of the measurement x k ) probability density of the parameters a k we have recurrent formulas for density p the first and second moments a, P p (a k x k, a k 1, a k 2,...)=p (a k x k, a k 1 )=p (x k a k ) p (a k, a k 1 )/p (x k ) P 1 k = P 1 + k k 1 H k R 1H k k P 1 k âk = P 1 k k 1âk k 1 + H k R 1x k k, â k k 1 =f k (â k 1 ) F k = f k(â k 1 ) a k 1 H k = h k(â k k 1 ) a k P k k 1 =F k P k 1 F k+γ k If measured parameter is a scalar u w k = w k /(1 + w k h kp k k 1 h k ) P k = P k k 1 w k (P k k 1 h k )(P k k 1 h k ) â k = â k k 1 + w k P k h k (u k h k (â k k 1 ) where w k = 1/σ 2 u k is the weight characteristic of measurement u k.

7 Least Squares and Recurrent Issues Selection of the state a, the dynamic model and the propagation method for state and covariances. Calculation of the initial approximaiton recurrent algorithm, â 0, P 0 for the The technique for calculating the noise matrix of the system Γ k for the recurrent algorithm. Calculation of the initial approximaiton a 0, for minimization of Ψ(a); Selection of technique for reaching the minimum of Ψ(a).

8 The state vector a a =(x, y, z, ẋ, ẏ, ż) for the certain time in the local rectangular coordinate frame (LRCF) related to geographical directions. The basic plane is the plane of local horizon, the x axis is in the plane of local horizon and is directed to the west, the y axis is directed normal to the plane of local horizon upwards ad z axis is in the plane of local horizon and is directed to the north. x = d sin α cos β y = d sin β z = d cos α cos β

9 The dynamic model d(t) d = µ r 3 r 2 ω d ω ( ω r) 3J 2µR 2 e 2r 5 ( r+2( r, ω 0 ) ω 0 5( r, ω ) 0) 2 r r 2 d =(x, y, z), d =(ẋ, ẏ, ż) r =(x b x,y b y,z b z ) Greenwich coordinats of SO r = (x b x ) 2 +(y b y ) 2 +(z b z ) 2 r 1 = X 2 g0 +Y 2 g0 r 0 = X 2 g0 +Y 2 g0 +Z2 g0 z = Z g0 /r 0 r 1 = r 1 /r 0 R = R e (1 α z 2 ) b x =0 b y = r 0 b z =2 α R z r 1 ω = ω e (ω 0x,ω 0y,ω 0z ) ω e = /s ω 0x =0 ω 0y = z + r 1 b z /r 0 ω 0z = r 1 z b z /r 0 α= µ= km 3 /s 2 J 2 = R e = km X g0, Y g0, Z g0 Greenwich coordinats of LRCF origin

10 The propagation method for state and covariances. For solving the system of differential equations d = f( d, d) use the numerical Runge-Cutta methos of the 4th order. a i =(k (i) 1 +2k (i) 2 +2k (i) 3 + k (i) 4 )/6 k (i) 1 =τ i f(a i ) k (i) 2 =τ i f(a i +0.5k (i) 1 ) k (i) 3 =τ i f(a i +0.5k (i) 2 ) k (i) 4 =τ i f(a i +k (i) 3 ) For the operator F k (a k 1 ) we will take the matrix F k with dimensions 6 6 with the following non-zero elements: f i,i =1 for i =1, 2,..., 6, f i,i+3 =τ k for i =1, 2, 3, where τ k = t k t k 1. F n = τ n τ n τ n

11 Calculation of the initial approximaiton â 0, P 0 for the recurrent algorithm, The state and covariance estimate is determined from the 1st two observations d 1 and d 2. The technique for calculating the noise matrix of the system Γ k for the recurrent algorithm. Γ 11,k = Γ k = ( Γ 11,k where 110 σd 2/(τ n 2 ( t ef,d) 4 ) σα 2 /(τ n 2 ( t ef,α) 4 ) σβ 2 /(τ n 2 ( t ef,β) 4 ) ) σ d,σ α,σ β are the RMS values of typical values of the errors of measured parameters, t ef,d, t ef,α, t ef,β efficient memory of the algorithm for parameters d, α, β (parameters of the algorithm selected on experimental basis).

12 Calculation of the initial approximaiton a 0, for minimization of Ψ(a); For the initial approximation a 0 we use the estimate â n, obtained using measurements x 1, x 2,..., x n by the recurrent algorithm. Selection of technique for reaching the minimum of Ψ(a). The major effect of updating by least squares is the better accuracy of the velocity components u=(ẋ, ẏ, ż) of the state vector a. The position parameters u=(x, y, z) of the vector a in fact are not updated. Thus we suggest to make minimization of Ψ(a) =Ψ(u, u) =ψ( u) only by the vector u. For the point of the minimum u min of the function ψ( u) for the case when it is reached by one iteration we suggest the Newton s formula ( ) 1 2 ψ u min = u 0 u 2(a 0) ψ u (a 0) where the first and the second derivatives of the function ψ( u) with respect to parameters ẋ, ẏ, ż are calculated by finite differences method.

13 New (Guarantee) Approach Fundamental problem with the traditional methods (joint and recurrent) is the assumption of uncorrelated measurements and known measurement statistics. Because there is no a priori orbit identifying bad measurements can be difficult. Basis of the Guarantee method is a guaranteed range of the measurement errors. Guarantee means that the algorithm produces not only the orbit estimate but the maximum possible errors in the estimated parameters. Leads to a more accurate estimate of the orbit and uncertainty. Provides an upper bound on the state errors. This feature hin- More computationally intensive. dered early use.

14 Let for the moments t k (t k t k+1 ; k = 1, 2,...n) we acquire measurements u k of certain functions h k (c) of the m-vector of parameters c (m<n), and the errors of the measurements δu k =u k h k (c) are limited from above by constant δ k,max. The limits for errors of the measurements define in the m-dimensional space of parameters c =(c 1, c 2,..., c m ) a domain of possible values of c D n = n {u k δ i,max h k (c) u k δ k,max } k=1 Project this domain to the coordinate of the components of vector c and among the projected points for each axis find the most right c n,r and the most left c n,l. They define the boundaries (maximum and minimum values) for the changes of each component of parameter c. In this case the estimate c n of parameter and maximum errors δ c n,max of this estimate are naturally defined as c n = 1 2 (c n,r + c n,l ) δ c n,max = 1 2 (c n,r c n,l ) When the measured parameters are linearly related to the determined parameter a, finding c n and δ c n,max is a linear programming problem.

15 The Model Example m =1 h(c) =c u k = c + δ k, δ k,max = δ max. D n = n {[ u k δ max,u k +δ max ]} = [ max k=1 c n = 1 2 (max k u k +min k u k ) k u k δ max, min k δ c n,max =δ max 1 2 (max k [ a, b ] - interval with left end a and right end b. u k +δ max ] u k min k u k ) c n does not depend on δ max If we know δ max precisely the value δ c n,max is a correct upper estimate for the error of the estimate c n, for any correlations of the errors of different measurements. For not correlated errors of measurements the estimate c n with regard to accuracy is not inferior and sometimes can be essentially more accurate (!) then the estimate n ĉ n = 1 n u k of the least squares technique. k=1 for uniform distribution: σ cn 1.4 δ max n for triangular distribution: σ cn 0.46 δ max n σĉn 0.58 δ max (1) n σĉn 0.41 δ max (2) n

16 The Real Algorithm Assume there are measurements range d k, azimuth α k and elevation β k at the times t k (k =1, 2,..., n). The error limits are defined be δ d,max,δ α,max,δ β,max. Let epoch be t=0.5 (t 1 +t n ) and orbital parameters are c =(d, α, β, d, α, β). Divide the measurements into n/2 groups. Each group defines two points. The k-th group is (t k, t 0.5n+k). From two points at two different times determine the state or orbit defined by č k =(ď, ˇα, ˇβ, d, ˇ ˇ α, ˇ β)k. The determination of the states č( t) assume 2-body motion. Now computer the measurements that would give this solution including J 2. Integrate the equation for the deviations of r, ṙ from Keplerian orbit with the initial conditions r ( t )= ṙ( t )=0 and r = r k to obtain the errors in the measurements r k (t k ). Then compute the corrected measurements r k =r k r k and transform the r k to the measurement frame. The domain D k is approximated by a six-dimensional parallelepiped with the center at č k and 64 apexes determined by č k ± δ d,max j 1 ± δ α,max j 2 ± δ β,max j 3 ± δ d,max j 4 ± δ α,max j 5 ± δ β,max j 6 where j i (i=1,2,...6) are the lines of the matrix of partial derivatives (c k ) J = (u k, u. The boundary projections c k,l and c k,r of these 0.5n+k) 64 points determine the boundaries of the vector interval (c k,l, c k,r). Then (c l, c r )= n/2 (c k,l, c k,r) c n = 1 2 (c n,r + c n,l ) δ c n,max = 1 2 (c n,r c n,l ) k=1

17 References 1. Bolshakov I.A., Repin V.G. Issues of non-linear filtering, Automatics and telemechanics, v.23, N.4, Shapiro I.I. The Prediction of Ballistic Missile Trajectories From Radar Observation, Massachusetts Institute of Technology Lincoln Laboratory, McGraw-Hill, New York-Toronto-London, Mudrov V.I., Method of least modules, Moscow, Kotov E.O., On the problem of determination of satellite orbit with limited errors of measurements, Space research, v. 19, issue 4, Moscow, 1981 (in Russian). 5. Cramer H., Methods of Mathematical Statistics, Princeton University Press. Princeton, New York, Z.N. Khutorovsky., Efficient memory of the discreet Kalman s filter, USSR Academy of Science, Automatics and Telemechanics, Nauka, Moscow, N.2, 1972, (in Russian). 7. Vallado D.A., Fundamentals of Astrodynamics and Applications, Second Edition, Microcosm Press,

18 2001.

19 SUMMARY Obtaining the best initial orbit is vital for correlating tracks and developing the future space catalog. Philosophy and approach US and Russian methods of initial orbit determination are different. Current Russian methods of initial orbit determination have been reviewed. New method called The Guarantee method has been developed. Provides a more accurate orbit estimate. Provides upper bound on estimated state error. More computationally intensive.